Robert
Neel

Research

My research interests are in geometric analysis and probability. I have been especially interested in the overlap of these two fields, using Brownian motion and diffusions as tools in geometry. Particular topics of interest include classical minimal surfaces, more general minimal submanifolds, mean curvature flow and Ricci flow for surfaces, potential theory on manifolds, random walks on Riemannian and sub-Riemannian manifolds, small-time heat kernel asymptotics on Riemannian and sub-Riemannian manifolds, martingale methods in analysis, and couplings of semi-martingales.

Below is a list of publications and preprints, most with links to the paper in some form.

Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group (with D. Barilari and U. Boscain), to appear in Ann. Fac. Sci. Toulouse Math.
Preprint from arXiv.
Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling (with A. Agrachev, U. Boscain, and L. Rizzi), to appear in ESAIM Control Optim. Calc. Var.
Preprint from arXiv.
Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry (with U. Boscain and L. Rizzi), Adv. Math. 314 (2017), 124-184.
Preprint from arXiv.
On the heat diffusion for generic Riemannian and sub-Riemannian structures (with D. Barilari, U. Boscain, and G. Charlot), Int. Math. Res. Not. IMRN (2017), no. 15, 4639-4672.
preprint from arXiv.
A stochastic target approach to Ricci flow on surfaces (with I. Popescu), Ann. Probab. 44 (2016), no. 2, 1341-1425.
e-print from Project Euclid (subscription required), or the preprint from arXiv.
Martingales arising from minimal submanifolds and other geometric contexts, Illinois J. Math. 58 (2014), no. 2, 323-357. [actually published in 2015]
e-print from Project Euclid (subscription required), or the preprint from arXiv.
Brownian motion and the parabolicity of minimal graphs.
Unpublished preprint from arXiv.
Brownian motion and the Dirichlet problem at infinity on two-dimensional Cartan-Hadamard manifolds, Potential Anal. 41 (2014), no. 2, 443-462.
e-print from Springer, or the preprint from arXiv.
On parabolicity and area growth of minimal surfaces, J. Geom. Anal. 23 (2013), no. 3, 1173-1188.
e-print from Springer, or the preprint from arXiv.
Small time heat kernel asymptotics at the sub-Riemannian cut locus (with D. Barilari and U. Boscain), J. Differential Geom. 92 (2012), no. 3, 373-416.
e-print from Project Euclid, or the preprint from arXiv.
Stochastic methods for minimal surfaces, Geometric analysis: partial differential equations and surfaces, Contemp. Math., vol. 570, Amer. Math. Soc., Providence, RI (2012), 111-136.
e-print, from AMS eBooks.
A martingale approach to minimal surfaces, J. Funct. Anal. 256 (2009), no. 8, 2440-2472.
e-print from ScienceDirect, or the preprint from arXiv.
The small-time asymptotics of the heat kernel at the cut locus, Comm. Anal. Geom. 15 (2007), no. 4, 845-890.
e-print, from International Press, or the preprint from arXiv.
Analysis of the cut locus via the heat kernel (with D. Stroock), Surveys in Differential Geometry, Vol. 9, International Press, Boston (2004), 337-349.
Equilibrium configurations for a floating drop (with A. Elcrat and D. Siegel), J. Math. Fluid Mech. 6 (2004), no. 4, 405-429.
e-print, from SpringerLink.
C-singular solutions of the capillary problem (with R. Finn), J. Reine Angew. Math. 512 (1999), 1-25.
e-print, from De Gruyter.